Best Known (57, 79, s)-Nets in Base 3
(57, 79, 192)-Net over F3 — Constructive and digital
Digital (57, 79, 192)-net over F3, using
- 31 times duplication [i] based on digital (56, 78, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 26, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 26, 64)-net over F27, using
(57, 79, 285)-Net over F3 — Digital
Digital (57, 79, 285)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(379, 285, F3, 22) (dual of [285, 206, 23]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0) [i] based on linear OA(373, 254, F3, 22) (dual of [254, 181, 23]-code), using
- construction XX applied to C1 = C([105,124]), C2 = C([103,122]), C3 = C1 + C2 = C([105,122]), and C∩ = C1 ∩ C2 = C([103,124]) [i] based on
- linear OA(366, 242, F3, 20) (dual of [242, 176, 21]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {105,106,…,124}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(366, 242, F3, 20) (dual of [242, 176, 21]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {103,104,…,122}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(371, 242, F3, 22) (dual of [242, 171, 23]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {103,104,…,124}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(361, 242, F3, 18) (dual of [242, 181, 19]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {105,106,…,122}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 6, F3, 1) (dual of [6, 5, 2]-code) (see above)
- construction XX applied to C1 = C([105,124]), C2 = C([103,122]), C3 = C1 + C2 = C([105,122]), and C∩ = C1 ∩ C2 = C([103,124]) [i] based on
- 25 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0) [i] based on linear OA(373, 254, F3, 22) (dual of [254, 181, 23]-code), using
(57, 79, 6544)-Net in Base 3 — Upper bound on s
There is no (57, 79, 6545)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 49 298435 391870 261198 585985 397301 520827 > 379 [i]